In these talks I’ll try to explain these classes of groups so peculiarly separated by chance. I’ll start in § 1 with amenable groups, which have been studied the longest (since von Neumann in the 1920s, who was investigating the Banach–Tarski paradox as Henry explained last term), and are a very natural class of groups to look at. Then I’ll move on in § 2 to an enormous superset of the amenable...

Explicit substitution calculi are extensions of the -calculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution c...

Black holes are probably the most fascinating objects in the Universe. They result from the attractive nature of gravity, which pulls any object towards any other object. A large ball of matter thus tends to compress under its own gravitational attraction. Remarkably, if this ball is large enough, the other fundamental forces – electromagnetic, weak and strong – are unable to resist the compres...